[1927] on the Calculation of the Spectroscopic Terms Derived From Equivalent Electrons

7/31/2019 [1927] on the Calculation of the Spectroscopic Terms Derived From Equivalent Electrons

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JUNE, 19Z7 PHYSICAL RE VIED' VOL UME Z9

ON THE CALCULATION OF THE SPECTROSCOPIC TERMSDERIVED FROM EQUIVALENT ELECTRONS

BY HENRY NoRRIs RUssELL

ABSTRACT

The calculation of the spectroscopic terms which result from an atomic con-figuration containing several equivalent electrons, in which Pauli s restriction isoperative, can be made very simply by the extension of a notation due to Breit. Theresults are in agreement with those previously given by Hund for P and d electronsand those of Gibbs, Wilber and %hite for f electrons. Some minor alterations innotation are suggested.

' 'T IS now mell established that the spectroscopic terms belonging to an atom'. in a given state of ionization depend upon the combined inHuence of allthe electrons outside the complete "shells. " When these electrons areunlike (differing in either their total or azimuthal quantum numbers) thedetermination of the terms produced by any configuration is very simple; butwhen they are equivalent in these respects, the restriction stated by Paulibecomes operative, according to which no two electrons in the same a'tom canhave the same values for all four of the quantum numbers which define theirstate. The following method permits a rapid analysis of the effects of thislimitation.The notation is substantially that of Hund's book. ' The state of u single

electron in an atom is completely defined by five quantum numbers, s, /, n,m, and m~. Of these, s defines the "spin" and is always ', (in the usual unitsof h/2z), / is less by a unit than Bohr's azimuthal quantum number, and n isequal to Bohr's total quantum number. The quantities m, and m& are magneticquantum numbers, giving the orientations of the spin-axis and the orbitplane in a (hypothetical) magnetic field strong enough to break down allcouplings between individual orbital or spin vectors; m, has the value+-while m~ runs from l to by steps of a unit. Pauli s restriction then demandsthat no two electrons in the same atom have the same values of n, 1, m, andm~. The small letters, s, p, d, f, , are used to describe electrons for whichl = 0, 1, 2, 3, ; so that, for example, 2s, 4p, 3d, have exactly the same mean-ings as Bohr's 2~, 4~, 33.

States of an atom (which correspond to the separate magnetic levels intowhich a component of a multiple term is divided) may be defined by thenumbers S, L, mq and mL, .' Of these, S represents the vector sum of the s's

Hund, Linienspektra und periodisches System der Elemente. Berlin, 1927. (JuliusSpringer) .

2 Hund uses l; for the electron and l for the atom; but if both are to be used as subscripts,as is described here, the capital letter appears preferable. The use of s (and S) to denote bothspin-vectors and orbits (or terms) is not likely to lead to any misunderstanding, and it is notworth while to change the notation already adopted by several active workers to avoid thisformal objection.

782


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SPECTROSCOPIC TERMS FROM EQUI VALENT ELECTRONS 783

and defines the multiplicity, being 0, 1, , for singlets, doublets, trip-lets, ; L is the vector sum of the l's and is 0, 1, 2, 3, , for S, I', D,F, . , terms. The magnetic quantum numbers, as ordinarily defined, arein weak fields m=nzl. +m8, and in strong fields m'=mr, +2nzq.When only one electron is active (as in Na I, Ca II, etc), the quantities

denoted by small and large letters become identical. In this case a total quan-tum number n can be assigned to the atomic state. When more than oneelectron is active, no such assignment is possible, and a complete descriptionof the situation demands the specification of n and / for each electron. Forexample, the lowest energy-state of Ti I is (3d)' (4s)', 'F~, two of the fouractive electrons being in 3d orbits, and the other two in 4s orbits.

2, 1 0)S

Z f 0

0'I

0 -1I-ZI

J= i E

-i!II PI

I

I 3 II I

Fig. 1. Relation between nzz, mz, and the inner quantum number j, for a regular 'D term.The relation first stated by Pauli, between m~, ml, and the inner quantum

number j' has been simply expressed graphically by Breit. ' The values ofml. are written above the top of a rectangle, those of nzq at the left, and themagnetic quantum numbers m = rnq+m J. inside it. The magnetic levels whichunite into a single component of the term in the absence of an external field,are then obtained by dividing the rectangle into L-shaped strips, as shown bythe dotted lines in Fig. 1, which corresponds to a 'D term (S=1, I =2). Thethree "runs" of I from 3 to , 2 to and 1 to correspond respectively

I

7I 2 II

'5

( Y

2 1 0 -1 -2

2 2 2 28 I i2 2

a j=4

Fig. 2. Relation between mj, mL, and the inner quantum number j, for an inverted F term.to the components 'D3, 'D2, 'DI. The maximum numerical value of m in each"run" gives the inner quantum number of the component, in Sommerfeld'snotation.The arrangement here given holds good in general for "regular" terms.

For inverted terms, the strips are inverted as shown in Fig. 2, representing a' j is not written as a capital (a) to avoid confusion with Lande's usage, (b) because

inner quantum numbers have a meaning only for atomic states, and not for the separateelectron-orbits.

4 Breit, Phys. Rev. 28, 334 (1926).


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HENRY NORRIS RUSSELL

'J" term. The same arrangement may be obtained by keeping the strips asbefore but changing the sign of each individual m, and ml.When two unlike electrons occur in an atom, the values of L and 5 resulting

from the combination of their individual values of l and s may be derived bythe aid of Sommerfeld's form of the vector-model, or as Breit has shown, bythe graphical process just illustrated. For example, Fig. 1 represents the com-bination of a P-electron (I=1) and a d-electron (l =2) provided that the outervertical and horizontal rows are supposed now to represent the two sets ofvalues of m& and the quantities inside the rectangle the values of m& whichare obtained by adding the others in all possible conbinations. Dividing theminto strips as before we obtain runs of m~ from 3 to , 2 to and 1 to 1,which give L = 3, 2, 1, or D, P, 5, terms. If additional electrons (not equivalentto any previously considered) are to be added, the values of nzrfor e,ach termof this first resultant are to be combined independently with those of m& forthe new electron.The s-vectors are similarly treated; thus Fig. 2 represents the addition of an

additional electron (s=-', ) to a configuration giving a sextet term S=3, thenew runs of ms being from 7/2 to 7/2 and 5/2 to /2, and the new termsseptets and quintets.So long as the electrons are all dissimilar there is no restriction on these

combinations. Any pair of values of m&, m& may be added to give a new m&and any other pair of values of m&, m, to give a new ms. It is therefore suffi-cient to consider the combinations of the two separately; and terms of anygiven type (5, P, D) will appear in both the multiplicities produced by add-ing the new electron.But when the electrons are equivalent in their total and azimuthal quantum

numbers, Pauli s restriction operates. All cases in which both m~ and m, arethe same for a pair of electrons must be excluded, and, what is more, casesobtainable from one another by a mere permutation of the order in which theelectrons are counted correspond to the same atomic configuration, and giveexactly the same energy-level.

How this works may best be seen by an illustration. In the case of twoequivalent p-electrons, the diagram for mz, is as follows.

0 [2]

0 1 [0] 0 [2]

When mq = 0, the individual values of m, must be + 2 and 2. The two elec-trons are dissimilar in this respect, and hence the combination of the m~ sis unrestricted, giving runs of mq, 2 to , 1 to and 0. But when ma=1(or ) the two m, 's are alike, and the m~'s are restricted. The values of mr.on the diagonal, which are bracketed above must now be excluded, and thevalues below the diagonal become mere duplicates of those above. All thatremains of m~ is the run 1 to ..


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SPECTROSCOPIC TERMS FROM EQUI VALENT ELECTRONS 785

Collecting these results we have,

1 to 2to , 1 to 1, 0

1 to 3gl

We have evidently here exactly the sets of values of m& and mL, which arerequired to form the terms indicated at the bottom.

In order to distinguish between primed and unprimed terms a very simplerule which is now being followed by several investigators may be deducedfrom Heisenberg's statement that, in any electron transition which gives riseto radiation, one electron changes its azimuthal quantum number by one unit,while at the same time one other may change by two. If then we assign tothe various types of orbits 0, 1, 2, 3, for s, p, d, f, electrons respectively (whichare Hund's values of l) it is evident that the sum of the l's, in any transition,must change from odd to even, or vice versa. This gives us two groups ofterms, ven terms (l sums even); S, P', D, P', G, etc. : odd terms (l sumsodd): S', P, O', P, G', etc.hich may be distinguished by the fact that th' eodd terms have an odd number of p and f electrons, taken together, in theconfiguration. If primed terms are defined in this way, a notation is obtainedwhich is consistent with the accepted notation for the sodium, calcium andaluminum groups, and which may be applied without ambiguity to all cases.In the case of three p-electrons, the sum ms = + 3/2 may be obtained only

when all three of the values of m, are alike. The three values of m~ must thenall be different; that is, they must be 1, 0 and 1, and ml, = 0. The sum ma =-',is obtained when two of the m, 's are -,and the other ~. The first two givethe run 1 to for mr, . Since the third electron is dissimilar (having a differentvalue of m. ) this run combines freely with the m; of the third electron, givingruns from 351. of 2 to , 1 to and 0. The case where m, =,is exactlysimilar.We thus find the array

my= + ~~+ 1 2 to 2, 1 to 1, 0

2P 4g~

Beyond this point we need not go, for, as is well known, a complete shellof six P-electrons must have mal=0, m~ , 0, giving a 'S term. A shell of fiveelectrons gives the same values as a single electron, and one of four the sameas one of two, except that the sign of each individual rnq and ml, is changed,whence it follows that the terms are the same as those previously calculated,but are inverted.The advantages of the present method of calculation are more apparent

in the case of equivalent d-electrons. When discussing it, we will, for brevity,write (4) to denote the "run" 4 to , etc. The free combination of any two


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runs (m) and (n) (where we may suppose m&n), then gives a set of runs(m+n), (m+e 1), and so on to (m s).For two d-electrons, we have the diagram

[4]1 3

0 2

1

0

1 0

3 2

[2l

[ol0 1.

[1

When rug=0 combination is unrestricted, but when my=+1 the diagonalvalues must be rejected and the quantities below the diagonal are duplicatesof those above, so that we have simply the runs (3) and (1). Our array thenbecomes

ms +1 (3) (1)= (4) (3) (2) (1) (O)

'G 'Ii ' 'D 'I" 'SWe may next note that, with 6ve d-electrons and m, = +5/2, all the m, 's

are of the same sign and mz, must necessarily be 0. With four electrons andnz, = +2, only one of the five possible values of m& is lacking in any set, sothen mr, has the run (2) as in the case of a single electron. Finally, for threeelectrons, and m, = + 3/2, two are lacking, and mr, has the runs (3) and (1).Very little further calculation is now necessary. For three electrons and

ms ',

, we have two with m, =+',

, giving runs (3) and (1), and one withm, =giving the run (2). These runs combine without restriction, giving,in the erst case, runs of (1), (2), (3), (4) and (5), and in the second (1), (2)and (3). Here we have

ms = + ', mr, = (1) (2) (3) (4) (5)+ 1

(1) (2) (3)(1) (3)

~D 4P' 'G 'H' 4I" 'D

For four electrons, mq= + 1 can be obtained from three electrons from whichm, has one sign and mr, = (3) or (1), and one of the other sign, giving the 'same combinations as before, while when m, q 0, m, =+-', for two and ,forthe other two, giving runs of (3) and (1) to be combined freely with another(3) and (1).In writing the resulting array, we may record simply the number of runs of

mI. of each length found for any given value of mq, as is done below. Thenumber of terms of the highest multiplicity and of any given sort (8, P, D)is then equal to the number of runs listed under the corresponding value ofmI. and in the row headed by the highest value of mq. The numbers of termsof any lower multiplicity are found by subtracting, from the numbers in the


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SPECTROSCOPIC TERMS FROM EQUI VALENT ELECTRONS

row headed by the corresponding value of nz8, the numbers in the row nextabove these.The results for four and 6ve equivalent d-electrons are shown in Table I.

In the latter case, ms = + 5/2 gives, the single run (0), ms = + 3/2 the resultantof runs of (2) for the four similar electrons, and (2) for the other one andms= +2 that of (I) and (3) for the group of three electrons and (1) s,nd (3)again for the group of two.

TABLE IThe array for four and five equivalent d electrons.

Fourelectrons

Five

mgm$= +2

+10

mg = +5/2+ 3/2+ 1/2

(o) (1) (2)1

2 22 2 4

(3) (4) (5) (6)

TermsFour Quintets

electrons TripletsSinglets

D

12

Fiveelectrons

SextetsQuartetsDoublets

12

our analysis is now complete. The resulting terms may be arranged in theform shown in Table II.

TABLE IIResulting terms for equivalent d electrons

d10

d,d'd2 d8d3 d7d4, d'd6

1S2D3P 3F~4PI4FI'D6S

1S . 1D 1G'D 'P' 'D 'F' 'G 'H''P ' 'F"P' 'D 'F' 'G 'H!4pr 4F~ ~ 4D . 4G

t t

1S 1D lG 1S 1D 1F 1G 1I2D 2pr 2D 2F~ 2G 2H~. 2S 2D 2F~ 2G 2I

This is identical, in content, with Hund s table' but brings out the note-worthy regularities in arrangement in a somewhat different manner. Theterms of lowest energy level are found in the second column and the otherterms of practical importance in the third.The case of equivalent f electrons demands a little more reckoning. Two

such electrons may be treated in the same fashion as two d-electrons, giving

(I) (3) (5)= (o) (&) (2) (3) (4) (5) (6)lg 3jP~ 1D 3P~ 16 3II~ 11

6 Hund, LiniensPektra, p. 119. The table ih his original paper, Jacks a 'G term for d4 anda 'S for d6.


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With three electrons, and ms=+3/2, the actual cases must once more becounted, and care must be taken not to count the same combination twice.This may be assured by combining the first two electrons as above and addinga third only when its value of ml, is less than for either of the other two. Inthis way each permutation will evidently be counted once and only once.For the two electrons the exhibit of the permissible values of m~ and uzi,

forms a triangle, as follows.

nz)=2 1 0

m1. =5 4 3

0 3

3 2 1 0 21 0 1 0

We may begin by combining M& = 5 in the first column of the trianglewith the values of nz~ in subsequent columns and in the outer row above, thenthe values M' r, 4, 3, in the next column with the numbers in the followingcolumns of the outer row and so on. Thus we get the series of numbers

6, 5, 4, 3, 2; 4, 3, 2,3, 2, 1, 0;

2, 1, 0;1, 0,0, 1, 2;

0, )2')3'. ~

which may be immediately rearranged into the runs (6), (4), (3), (2), (0).For ms + ~ we have two electrons with m, of like sign, giving the runs (1),

(3), (5), and one of opposite sign with the iun (3). The final array may thenbe written as follows.

mr,

528=+ j+1

Terms

(0)1

(2)1

3D'

(3)1

3

(4)

1

G'

(5) (6) (7) (g)

Quartets

Doublets 2 2 1 1

The remaining computations are now straightforward. Four electrons withrr4 alike give again the runs (0), (2), (3), (4), (6), five give (1), (3), (5), sixgive (3) and seven give (0).The results were worked out independently by Messrs. Gibbs, Wilber and.

White, and may be found in their paper, (which immediately follows this).


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SPECTROSCOPIC TERMS FROM EQUIVALENT ELECTRONS 789

(By mutual agreement, the theory has been presented by one author, and thenumerical results, so far as they are new, by the others. )A word may be said about the notation of the terms corresponding to high

values of L. The letter J has been omitted by Hund in accordance with Germanusage. P and S are preoccupied. The list of letters then becomes

I.= 0 1 2 3 4 5 6 7 8 9 10S P D Ii 6 H' I X L M

L= ii 12 13 14 15 16 17 18 19 200 Q R T L7 V W X Y Z

Even this extension is barely adequate to include what may be anticipatedamong the rare earths. The highest multiplicity to be expected is 11,which, asHund points out' should occur in Gd only, and give a term "Ii arising from theconfiguration f'd's, which must be either the normal state or a very low metas-table one. The highest value of I. among the "middle terms" with one excitedelectron should occur in the same spectrum, arising from the configuration(4f) ' (5d)' (6p), which should give a term for which mr, = 16, of type ' V'. Similarterms originating in the configuration f'd'p or f'd'p should occur in Eu and Tb.Still greater values of L could be reached in highly excited states; for examplethe configuration (4f)'(5f) (5d) (6d) (6p) representing an atom of Tb withthree excited electrons, should give a maximum value L=20, and a termwhich would demand the notation 'Z.

Among the multitude of levels given by this configuration mould also besome derived from the 'S term of origin f', which would be of multiplicity 12,and all types from "Sto "I.. These terms however would be very unlikely togive lines strong enough to be observable.In conclusion, reference may be made to the beautiful manner in which

Breit's graphical process4 solves the problem of the limits of series in complexatoms, and shows which components of a given term in the arc spectrum(for example) go to given components of the limiting term in the spark spec-trum. The results are naturally in accordance with those given by Hund inFig. 34 at the end of his book.

PRINCETON UNIVERSITY)OBSERVATORY,

April 4, 1927.

' Hund, Linienspektru, p. 177.

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[1927] on the Calculation of the Spectroscopic Terms Derived From Equivalent Electrons